We give a new “nested adds ” circuit for implementing Shor’s algorithm in linear width and quadratic depth on a nearest-neighbor machine. Our circuit combines Draper’s transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another version, with slightly larger depth, using only reversible classical gates. We do not know which version will ultimately be cheaper to implement. 1

We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level of ^ ` = (6-2p2) /7 ss 45%, thereby improving on a previous boundof 50 % (due to Razborov [18]). More precisely, the circuit model for which we prove this bound contains perfect gatesfrom the Clifford group (CNOT, Hadamard, S, X, Y, Z)and arbitrary additional one-qubit gates that are subject to

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Abstract. Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for QIP on constant depth circuits containing the unbounded fan-out gate. These results are shown by reducing a QIP-complete problem to a logarithmic depth version of itself using a parallelization technique. 1.

We propose and discuss two postulates on the nature of errors in highly correlated noisy physical stochastic systems. The first postulate asserts that errors for a pair of substantially correlated elements are themselves substantially correlated. The second postulate asserts that in a noisy system with many highly correlated elements there will be a strong effect of error synchronization. These postulates appear to be damaging for quantum computers.

We propose and discuss two conjectures on the nature of informa-tion leaks (decoherence) for quantum computers. These conjectures, if (or when) they hold, are damaging for quantum error-correction as required by fault-tolerant quantum computation. The first conjecture asserts that information leaks for a pair of substantially entangled qubits are themselves substantially positively correlated. The second conjecture asserts that in a noisy quantum computer with highly entangled qubits there will be a strong effect of error synchronization. We present a more general conjecture for arbitrary noisy quantum systems: prescribing (or describing) noisy quantum systems at a state ρ is subject to error E which “tends to commute ” with every unitary operator that stabilizes ρ.

Abstract. We consider a quantum polynomial-time algorithm which solves the discrete logarithm problem for points on elliptic curves over GF(2 m). We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation is O(m 2), which is an improvement over the previous bound of O(m 3). 1

We evaluate the performance of quantum arithmetic algorithms run on a distributed quantum computer (a quantum multicomputer). We vary the node capacity and I/O capabilities, and the network topology. The tradeoff of choosing between gates executed remotely, through “teleported gates ” on entangled pairs of qubits (telegate), versus exchanging the relevant qubits via quantum teleportation, then executing the algorithm using local gates (teledata), is examined. We show that the teledata approach performs better, and that carry-ripple adders perform well when the teleportation block is decomposed so that the key quantum operations can be parallelized. A node size of only a few logical qubits performs adequately provided that the nodes have two transceiver qubits. A linear network topology performs acceptably for a broad range of system sizes and performance parameters. We therefore recommend pursuing small, high-I/O bandwidth nodes and a simple network. Such a machine will run Shor’s algorithm for factoring large numbers efficiently.

This thesis presents the semantics of quantum stacks and a functional quantum pro-gramming language, L-QPL. An operational semantics for L-QPL based on quantum stacks in the form of a term logic is developed and used as an interpretation of quan-tum circuits. The operational semantics is then extended to handle recursion and alge-braic datatypes. Recursion and datatypes are not concepts found in quantum circuits, but both are generally required for modern programming languages. The language L-QPL is introduced in a discussion and example format. Various example programs using both classical and quantum algorithms are used to illustrate features of the language. Details of the language, including handling of qubits, general data types and classical data are covered. The quantum stack machine is then presented. Supporting data for operation of the machine are introduced and the transitions induced by the machine’s instructions are given.

In this paper we consider adversarial noise models that will fail quantum error correction and fault-tolerant quantum computation. We describe known results regarding high-rate noise, sequential computation, and reversible noisy computation. We continue by discussing highly correlated noise and the “boundary, ” in terms of correlation of errors, of the “threshold theorem. ” Next, we draw a picture of adversarial forms of noise called (collectively) “detrimental noise.” Detrimental noise is modeled after familiar properties of noise propagation. However, it can have various causes. We start by pointing out the difference between detrimental noise and standard noise models for two qubits and proceed to a discussion of highly entangled states, the rate of noise, and general noisy quantum systems. Research supported in part by an NSF grant, an ISF grant, and a BSF grant.